Jordan homomorphisms of associative algebras with involution
For an associative ring R with an involution *: \(R\to R\), let \(R^{(+)}\) be the associated Jordan ring and \(H(R,*)=\{\tau \in R|\) \(\tau^*=\tau \}\) the set of symmetric elements (a subring of \(R^{(+)})\). The author investigates to what extent the Jordan ring H(R,*) determines the associative ring R. An associative algebra R (over a commutative, associative ring containing \()\) with involution *: \(R\to R\) is involutively prime (semiprime) if for any two nonzero *-invariant ideals, \(I_ 1\), \(I_ 2\), \(I_ 1I_ 2=0\) (if it contains no nonzero *-invariant nilpotent ideals). For a subset A of R, denote by \(\) the subalgebra of R generated by A. The author's main results are the following two main theorems. Theorem 1. For associative algebras R and S with involutions \(\tau_ 1: R\to R\) and \(\tau_ 2: S\to S\), and S involutively prime, any Jordan epimorphism \(\phi\) : H(R,\(\tau_ 1)\to H(S,\tau_ 2)\) can be extended to a homomorphism of the algebra \(<H(R,\tau_ 1)>\) into S, or S is a central order in a simple algebra, 16 dimensional over its centre and \(\tau_ 2\) is an involution of symplectic type. - Theorem 2. For an associative algebra R with involution *: \(R\to R\), where R is involutively prime, any Jordan derivation d: H(R,*)\(\to R\) can be extended to a derivation of \(<H(R,*)>\) into R, or R is a central order in a simple algebra which is 16 dimensional over its centre and * is an involution of symplectic type.
scientific article; zbMATH DE number 4142242
scientific article